// Make newform 405.4.e.l in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_405_e();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_405_e_Hecke();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_405_4_e_l();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_405_4_e_l();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function ConvertToHeckeField(input: pass_field := false, Kf := []) if not pass_field then poly := [1, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]]; inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf); return Eltseq(inp_vec); end function; // To make the character of type GrpDrchElt, type "MakeCharacter_405_e();" function MakeCharacter_405_e() N := 405; order := 3; char_gens := [326, 82]; v := [2, 3]; // chi(gens[i]) = zeta^v[i] assert UnitGenerators(DirichletGroup(N)) eq char_gens; F := CyclotomicField(order); chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]); return MinimalBaseRingCharacter(chi); end function; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_405_e_Hecke();" function MakeCharacter_405_e_Hecke(Kf) N := 405; order := 3; char_gens := [326, 82]; char_values := [[0, -1], [1, 0]]; assert UnitGenerators(DirichletGroup(N)) eq char_gens; values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field F := Universe(values);// the Hecke field chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values); return chi; end function; function ExtendMultiplicatively(weight, aps, character) prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec primes := PrimesUpTo(prec); prime_powers := primes; assert #primes eq #aps; log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1) F := Universe(aps); FXY := PolynomialRing(F, 2); // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ... R := PowerSeriesRing(FXY : Precision := log_prec + 1); recursion := Coefficients(1/(1 - X*T + Y*T^2)); coeffs := [F!0: i in [1..(prec+1)]]; coeffs[1] := 1; //a_1 for i := 1 to #primes do p := primes[i]; coeffs[p] := aps[i]; b := p^(weight - 1) * F!character(p); r := 2; p_power := p * p; //deals with powers of p while p_power le prec do Append(~prime_powers, p_power); coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]); p_power *:= p; r +:= 1; end while; end for; Sort(~prime_powers); for pp in prime_powers do for k := 1 to Floor(prec/pp) do if GCD(k, pp) eq 1 then coeffs[pp*k] := coeffs[pp]*coeffs[k]; end if; end for; end for; return coeffs; end function; function qexpCoeffs() // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" weight := 4; raw_aps := [[4, -4], [0, 0], [0, 5], [-6, 6], [-32, 32], [0, 38], [26, 0], [100, 0], [0, 78], [50, -50], [0, 108], [266, 0], [0, -22], [-442, 442], [514, -514], [2, 0], [0, -500], [518, -518], [0, -126], [412, 0], [-878, 0], [-600, 600], [-282, 282], [-150, 0], [-386, 386], [-702, 702], [0, 598], [-1194, 0], [-550, 0], [0, -1562], [1846, 0], [0, 2208], [2334, -2334], [0, 700], [0, -2050], [-1852, 1852], [0, 2494], [2762, 0], [0, -3126], [78, -78], [-1300, 0], [1742, 0], [-3772, 3772], [0, 358], [-2214, 0], [-2600, 0], [0, 1168], [6478, -6478], [-646, 646], [0, -3750], [1482, 0], [0, -1400], [-3022, 3022], [-1248, 0], [0, -2106], [3638, -3638], [-6550, 0], [-4388, 0], [-546, 546], [6858, -6858], [0, -9282], [0, -4842], [-2594, 0], [0, -7332], [-1562, 1562], [-1426, 1426], [4008, -4008], [0, -8866], [0, 1714], [-1150, 1150], [4398, -4398], [1800, 0], [5874, -5874], [0, 2078], [7900, 0], [0, 7518], [1950, -1950], [13786, 0], [0, -6402], [0, -11150], [0, 13700], [5438, -5438], [7692, 0], [-1118, 0], [2600, -2600], [11958, -11958], [-17050, 0], [9494, -9494], [11418, -11418], [0, -7962], [6526, 0], [-17400, 17400], [1166, 0], [0, -7072], [0, -100], [2602, 0], [0, -11150], [-3638, 0], [-2078, 0], [5622, 0], [-16486, 16486], [11706, 0], [0, 25038], [-17550, 17550], [0, -10712], [-13654, 0], [-14166, 14166], [17842, 0], [0, 17600], [-27302, 27302], [0, 3794], [-13238, 0], [0, 11574], [-8300, 8300], [-7508, 0], [27378, -27378], [0, -1842], [-10114, 0], [0, -10402], [-7100, 7100], [0, 7118], [31278, -31278], [30054, -30054], [-4518, 0], [-29272, 29272], [-5798, 0], [-8950, 8950], [7800, 0], [8554, -8554], [0, -2882], [18700, 0], [0, -12242], [0, 31148], [-7694, 0], [0, 4518], [0, 39550], [22122, 0], [0, 16634], [0, -27586], [3850, 0], [10032, 0], [-20562, 20562], [0, -10322], [8846, 0], [-25350, 0], [-46000, 46000], [16998, -16998], [26494, -26494], [0, 21500], [25762, 0], [0, -30546], [32942, 0], [-27118, 0], [0, 38634], [1794, -1794], [-41732, 41732], [29200, 0], [48650, -48650], [-11334, 0], [0, 31178], [-4686, 4686], [-598, 0], [0, -41726], [24312, 0], [0, -40946], [-42282, 42282], [1172, 0], [31614, -31614]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_405_e_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_405_4_e_l();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. function MakeNewformModFrm_405_4_e_l(:prec:=2) chi := MakeCharacter_405_e(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 4)); S := BaseChange(S, Kf); maxprec := NextPrime(997) - 1; while true do trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]); B := Basis(S, prec + 1); S_basismat := Matrix([AbsEltseq(g): g in B]); if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then S_basismat := ChangeRing(S_basismat,Kf); f_lincom := Solution(S_basismat,trunc_vec); f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]]; return f; end if; error if prec eq maxprec, "Unable to distinguish newform within newspace"; prec := Min(Ceiling(1.25 * prec), maxprec); end while; end function; // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_405_4_e_l();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function MakeNewformModSym_405_4_e_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_405_e(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<2,R![16, -4, 1]>,<7,R![36, 6, 1]>],Snew); return Vf; end function;